Question

Prove that If x and y are both odd positive integers then x^2 and y^2 is even but not divisible by 4.

Answer 1

We know that any positive odd integers are of the form 4q+1 & 4q+3.
So let x = 4q+1 & y = 4q+3

Now,
x^2+y^2
= (4q+1)^2+(4q+3)^2
= 16q^2 +8q+1+9+16q^2 + 24q
= 32q^2 + 32q + 10
= 32q^2 + 32q + 8 +2
= 4(8q^2 + 8q + 2) +2
Now when we divide it by 4, there is a remainder of 2, so it is divisible by 2 and is even but it is not divisible by 4 because there remains a remainder of 2.